metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C18, D7⋊C9, C3.F7, C21.C6, C7⋊C9⋊C2, (C3×D7).C3, SmallGroup(126,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C18 |
Generators and relations for C7⋊C18
G = < a,b | a7=b18=1, bab-1=a3 >
Character table of C7⋊C18
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7 | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | 21A | 21B | |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | ζ98 | ζ97 | ζ92 | ζ9 | ζ94 | ζ95 | -ζ92 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | ζ3 | ζ32 | linear of order 18 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ94 | ζ98 | ζ9 | ζ95 | ζ92 | ζ97 | ζ9 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ32 | ζ3 | linear of order 9 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | ζ95 | ζ9 | ζ98 | ζ94 | ζ97 | ζ92 | -ζ98 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | ζ3 | ζ32 | linear of order 18 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | ζ94 | ζ98 | ζ9 | ζ95 | ζ92 | ζ97 | -ζ9 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | ζ32 | ζ3 | linear of order 18 |
| ρ11 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ92 | ζ94 | ζ95 | ζ97 | ζ9 | ζ98 | ζ95 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ3 | ζ32 | linear of order 9 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | ζ9 | ζ92 | ζ97 | ζ98 | ζ95 | ζ94 | -ζ97 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | ζ32 | ζ3 | linear of order 18 |
| ρ13 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ9 | ζ92 | ζ97 | ζ98 | ζ95 | ζ94 | ζ97 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ32 | ζ3 | linear of order 9 |
| ρ14 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | ζ92 | ζ94 | ζ95 | ζ97 | ζ9 | ζ98 | -ζ95 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | ζ3 | ζ32 | linear of order 18 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ98 | ζ97 | ζ92 | ζ9 | ζ94 | ζ95 | ζ92 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ3 | ζ32 | linear of order 9 |
| ρ16 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ95 | ζ9 | ζ98 | ζ94 | ζ97 | ζ92 | ζ98 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ3 | ζ32 | linear of order 9 |
| ρ17 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | ζ97 | ζ95 | ζ94 | ζ92 | ζ98 | ζ9 | -ζ94 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | ζ32 | ζ3 | linear of order 18 |
| ρ18 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ97 | ζ95 | ζ94 | ζ92 | ζ98 | ζ9 | ζ94 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ32 | ζ3 | linear of order 9 |
| ρ19 | 6 | 0 | 6 | 6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F7 |
| ρ20 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex faithful, Schur index 3 |
| ρ21 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex faithful, Schur index 3 |
►On 63 points
Generators in S63
(1 23 49 39 30 58 14)(2 40 15 50 59 24 31)(3 51 32 16 25 41 60)(4 17 61 33 42 52 26)(5 34 27 62 53 18 43)(6 63 44 10 19 35 54)(7 11 55 45 36 46 20)(8 28 21 56 47 12 37)(9 57 38 22 13 29 48)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63)
G:=sub<Sym(63)| (1,23,49,39,30,58,14)(2,40,15,50,59,24,31)(3,51,32,16,25,41,60)(4,17,61,33,42,52,26)(5,34,27,62,53,18,43)(6,63,44,10,19,35,54)(7,11,55,45,36,46,20)(8,28,21,56,47,12,37)(9,57,38,22,13,29,48), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)>;
G:=Group( (1,23,49,39,30,58,14)(2,40,15,50,59,24,31)(3,51,32,16,25,41,60)(4,17,61,33,42,52,26)(5,34,27,62,53,18,43)(6,63,44,10,19,35,54)(7,11,55,45,36,46,20)(8,28,21,56,47,12,37)(9,57,38,22,13,29,48), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63) );
G=PermutationGroup([[(1,23,49,39,30,58,14),(2,40,15,50,59,24,31),(3,51,32,16,25,41,60),(4,17,61,33,42,52,26),(5,34,27,62,53,18,43),(6,63,44,10,19,35,54),(7,11,55,45,36,46,20),(8,28,21,56,47,12,37),(9,57,38,22,13,29,48)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)]])
C7⋊C18 is a maximal subgroup of
C9×F7 C9⋊3F7 C9⋊4F7 C32.F7 D21⋊C9
C7⋊C18 is a maximal quotient of C7⋊C36 C7⋊C54 D21⋊C9
Matrix representation of C7⋊C18 ►in GL7(𝔽127)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 126 | 126 | 126 | 126 | 126 | 126 |
| 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 29 | 0 | 105 | 76 | 76 | 105 |
| 0 | 0 | 29 | 51 | 80 | 51 | 29 |
| 0 | 98 | 76 | 47 | 47 | 76 | 98 |
| 0 | 29 | 51 | 80 | 51 | 29 | 0 |
| 0 | 105 | 76 | 76 | 105 | 0 | 29 |
| 0 | 22 | 51 | 22 | 0 | 98 | 98 |
G:=sub<GL(7,GF(127))| [1,0,0,0,0,0,0,0,0,0,0,0,0,126,0,1,0,0,0,0,126,0,0,1,0,0,0,126,0,0,0,1,0,0,126,0,0,0,0,1,0,126,0,0,0,0,0,1,126],[24,0,0,0,0,0,0,0,29,0,98,29,105,22,0,0,29,76,51,76,51,0,105,51,47,80,76,22,0,76,80,47,51,105,0,0,76,51,76,29,0,98,0,105,29,98,0,29,98] >;
C7⋊C18 in GAP, Magma, Sage, TeX
C_7\rtimes C_{18} % in TeX
G:=Group("C7:C18"); // GroupNames label
G:=SmallGroup(126,1);
// by ID
G=gap.SmallGroup(126,1);
# by ID
G:=PCGroup([4,-2,-3,-3,-7,29,1731,583]);
// Polycyclic
G:=Group<a,b|a^7=b^18=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C7⋊C18 in TeX
Character table of C7⋊C18 in TeX